# Calculate integer summation when lower bound is a variable

How do you calculate the following sum?

\begin{equation*} \sum_{i=k}^n i \end{equation*}

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## 3 Answers

$\sum_{i=k}^n i = \sum_{i=0}^n i - \sum_{i=0}^{k-1} i = \frac{1}{2} \left[ n(n+1) - (k-1)k \right]$

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This is a special case for the formula for the sum of n terms in an arithmetic progression:

http://en.wikipedia.org/wiki/Arithmetic_progression

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Thanks. I probably should have specified that I was looking for a direct formula or derivation. – credford Aug 13 '10 at 22:01